Simultaneous Equations ① Elimination Method
Learn the “elimination method,” in which one variable is eliminated by adding or subtracting two equations.
Steps for the Elimination Method
- Make the absolute values of the coefficients of the variable you want to eliminate equal (multiply an equation by a suitable number if necessary)
- Add or subtract the two equations to eliminate one variable
- Find the value of the remaining variable
- Substitute the value found into one of the original equations to find the other variable
Example 1 (Coefficients are already equal)
{x＋y＝5 … ①
{x－y＝1 … ②
Solution:
①＋②: 2x＝6 → x＝3
Substitute into ①: 3＋y＝5 → y＝2
x＝3, y＝2
{x＋y＝5 … ①
{x－y＝1 … ②
Solution:
①＋②: 2x＝6 → x＝3
Substitute into ①: 3＋y＝5 → y＝2
x＝3, y＝2
Example 2 (Making the coefficients equal)
{2x＋3y＝12 … ①
{3x－y＝7 … ②
Solution:
②×3: 9x－3y＝21 … ②'
①＋②': 11x＝33 → x＝3
Substitute into ②: 9－y＝7 → y＝2
x＝3, y＝2
{2x＋3y＝12 … ①
{3x－y＝7 … ②
Solution:
②×3: 9x－3y＝21 … ②'
①＋②': 11x＝33 → x＝3
Substitute into ②: 9－y＝7 → y＝2
x＝3, y＝2
Key Points
- Make the absolute values of the coefficients of the variable you want to eliminate equal
- If the signs of the coefficients are the same, subtract; if they are different, add
- Substitute the value found into the “simpler equation”
Practice Problems
- {3x＋y＝11 {x－2y＝－2 Solve using the elimination method.
- {4x－3y＝5 {2x＋y＝7 Solve using the elimination method.
Answers and Explanations
- Answer:x＝20/7, y＝17/7
Explanation: ①3x＋y＝11, ②x－2y＝－2. ①×2: 6x＋2y＝22. ①×2＋②: 7x＝20, so x＝20/7. Substitute into ① to get y＝11－60/7＝17/7. Check: 3×20/7＋17/7＝11, 20/7－2×17/7＝－2. - Answer:x＝13/5, y＝9/5
Explanation: ①4x－3y＝5, ②2x＋y＝7. ②×3: 6x＋3y＝21. ①＋②×3: 10x＝26, so x＝13/5. Substitute into ② to get y＝7－26/5＝9/5. Check: 4×13/5－3×9/5＝5, 2×13/5＋9/5＝7.